Problem: Simplify the following expression: $y = \dfrac{-5x^2+26x- 24}{-5x + 6}$
Answer: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-5)}{(-24)} &=& 120 \\ {a} + {b} &=& &=& {26} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $120$ and add them together. The factors that add up to ${26}$ will be your ${a}$ and ${b}$ When ${a}$ is ${6}$ and ${b}$ is ${20}$ $ \begin{eqnarray} {ab} &=& ({6})({20}) &=& 120 \\ {a} + {b} &=& {6} + {20} &=& 26 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-5}x^2 +{6}x) + ({20}x {-24}) $ Factor out the common factors: $ x(-5x + 6) - 4(-5x + 6)$ Now factor out $(-5x + 6)$ $ (-5x + 6)(x - 4)$ The original expression can therefore be written: $ \dfrac{(-5x + 6)(x - 4)}{-5x + 6}$ We are dividing by $-5x + 6$ , so $-5x + 6 \neq 0$ Therefore, $x \neq \frac{6}{5}$ This leaves us with $x - 4; x \neq \frac{6}{5}$.